Residual Structure Theory in Modal Collapse Systems

This paper introduces a formal theory of residual structure inmodal collapse systems, focusing on one conjecture and two theorems:residual boundedness (conjecture), entropy attractor behavior (theo-rem), and spectral mode convergence (theorem). The modal collapsefunction ϕ(x) generates structural approximations, while the resid-ual field δ(x) quantifies deviations between empirical and predicteddensity. This work advances the Unified Resonant Structure Formula(URSF), linking residual feedback in cognitive geometry with formalstructural dynamics in modal systems, making residual intelligencemathematically operational.